Circular Orbit Velocity & Period
Speed and period of a circular Earth orbit from altitude.
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The engineering
A circular orbit is falling that keeps missing: gravity supplies exactly the centripetal acceleration, giving v = √(µ/r). The counterintuitive part is the sign — *higher* orbits are *slower*: ISS at 400 km does 7.67 km/s and laps Earth in 92 minutes, while GEO at 35,786 km ambles at 3.07 km/s taking precisely one sidereal day, which is the entire business model of communications satellites.
The period formula is Kepler's third law in SI clothing, and it's merciless about parking spots: only one altitude is geostationary, only certain altitudes give repeat ground tracks, and constellation designers pick radii the way watchmakers pick gears.
Where this math comes from
Kepler extracted the pattern from Tycho Brahe's naked-eye data — ellipses in 1609, the period-cube law in 1619 — and Newton's Principia (1687) showed both fall out of inverse-square gravity, complete with the cannonball thought-experiment that *is* this card. The math sat fully formed for 270 years waiting on propulsion.
Arthur C. Clarke priced out the 24-hour orbit for radio relays in a 1945 magazine article; Sputnik's 96-minute beep in 1957 turned v = √(µ/r) into headline news; and Syncom 3 parked at Clarke's altitude in 1964. Today µ_Earth is measured to nine digits by tracking the satellites this formula put there.
- 1619Johannes KeplerThird law: T² ∝ r³.
- 1687Isaac NewtonOrbits derived from gravitation; the cannonball argument.
- 1945Arthur C. ClarkeGeostationary orbit proposed for communications.
- 1957Sergei Korolev / Sputnik 1First artificial satellite proves the arithmetic.
- 1964NASA / Hughes Syncom 3First geostationary satellite.
See the full timeline of the math behind every calculator →
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